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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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NASA NTRS1995

Interaction of an interplanetary shock with the heliospheric plasma sheet

Interplanetary shocks often propagate along the heliospheric plasma sheet (HPS) where the interplanetary magnetic field (IMF) changes its polarity. This problem is investigated by the time-dependent 2.5-D MHD numerical model in the meridional plane. An example of computation is shown in the figure using density (log) contours and IMF vectors. Values of plasma parameters along the HPS fluctuate in time due to the Kelvin-Helmholtz instability. The HPS with its decreased intensity of the IMF as well as with its increased mass density causes a dimple in the shock structure (relatively weak for the forward shock, significant for the reverse shock, and very large for the contact discontinuity). Beyond the forward shock, the HPS is slightly compressed due to the post-shock increase of the azimuthal IMF component. Then follows expansion of the HPS surrounded by the highly-deformed contact discontinuity. A significant draping of IMF lines occurs around this structure that increases the meridional component of the IMF. This can cause a favorable condition for initiation of a geomagnetic storm.

Odstrcil, D.↗
NASA NTRS1992

Hydrodynamic instabilities in supernova remnants - Self-similar driven waves

An initial study aimed at elucidating the multidimensional aspects of the hydrodynamic instabilities in supernova remnants is presented. Self-similar solutions are found to exist for the interaction of a steep power-law density profile expanding into a relatively flat stationary power-law density profile. Consideration of the pressure and entropy profiles in the shocked 1D flows shows that the flows are subject to convective instability, by a local criterion. The growth rate for the instability becomes very large near the contact discontinuity between the two shocked regions. A linear analysis of the complete self-similar solutions shows that the solutions are unstable above a critical wavenumber and that the growth rate is greatest at the position of the contact discontinuity. The X-ray image of the remnant of SN 1572 (Tycho) shows emission from clumps of supernova ejecta, which is good evidence for instabilities in this remnant.

Chevalier, Roger A.↗
NASA NTRS1994

Application of the space-time conservation element and solution element method to shock-tube problem

An Euler solver based on the method of space-time conservation element and solution element is in this paper to simulate shock-tube flows involving shock waves, contact discontinuities, expansion waves and their intersections. Seven test problems are considered to examine the capability of this method. The numerical results, when compared with exact solutions and/or numerical solutions by other methods, indicate that the present method can accurately resolve strong shock and contact discontinuities without using any ad hoc techniques which are used only at the neighborhood of a discontinuity.

Wang, Xiao-Yen↗
NASA NTRS1993

Accurate upwind methods for the Euler equations

A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate, and can be considered as extensions of Godunov's scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer's MUSCL scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the 'smooth' part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new methods are applied in a manner slightly different from those in the literature. These methods are derived by approximating the Euler equations via linearization and diagonalization. At a 'smooth' interface, Harten, Lax, and Van Leer's one intermediate state model is employed. A modification for this model that can resolve contact discontinuities is presented. Near a discontinuity, either this modified model or a more accurate one, namely, Roe's flux-difference splitting. is used. The current presentation of Roe's method, via the conceptually simple flux-vector splitting, not only establishes a connection between the two splittings, but also leads to an admissibility correction with no conditional statement, and an efficient approximation to Osher's approximate Riemann solver. These reconstruction and upwind steps result in schemes that are uniformly second-order accurate and economical at smooth regions, and yield high resolution at discontinuities.

Huynh, Hung T.↗
NASA NTRS1995

Numerical magetohydrodynamics in astronphysics: Algorithm and tests for one-dimensional flow`

We describe a numerical code to solve the equations for ideal magnetohydrodynamics (MHD). It is based on an explicit finite difference scheme on an Eulerian grid, called the total variation diminishing (TVD) scheme, which is a second-order-accurate extension of the Roe-type upwind scheme. We also describe a non-linear Riemann solver for ideal MHD, which includes rarefractions as well as shocks. The numerical code and the Riemann solver have been used to test each other. Extensive tests encompassing all the possible ideal MHD structures with planar symmetries (i.e., one-dimensional flows) are presented. These include those for which the field structure is two dimensional (i.e., those flows often called '1 + 1/2 dimensional') as well as those for which the magnetic field plane rotates (i.e., those flows often called '1 + 1/2 + 1/2 dimensional'). Results indicate that the code can resolve strong fast, slow, and magnetosonic shocks within two to four cells, but more cells are required if shocks become weak. With proper steepening, we could resolve rotational discontinuities within three to five cells. However, without successful implementation of steepening, contact discontinuities are resolved with approximately 10 cells and tangential discountinuities are resolved with approximately 15 cells. Out tests confirm that slow compound structures with tow-dimensional magnetic fields are composed of intermediate shocks (so-called 2-4 intermediate shocks) followed by slow rarefaction waves. Finally, tests demostrate that in two-dimensional magnetohydrodynamics, fast compound structures, which are composed of intermediate shocks (so-called 1-3 intermediate shocks) preceeded by fast rarefaction waves, are also possible.

Ryu, Dongsu↗
NASA NTRS1990

The merging of quasiperpendicular collisionless shocks

The overtaking of one collisionless shock by another is studied by means of hybrid numerical simulations. The two shocks merge into a stronger shock and trailing nonshock discontinuities. The strong shock continues to propagate in the same direction as the two weaker shocks. The merging is shown to occur by a self-consistent process involving the interaction of ions reflected at the overtaking shock with the plasma upstream of the leading shock. The characteristic time scale for the merging is typically 1/Omega(i), where Omega(i) is the ion gyrofrequency. For exactly perpendicular shocks, the trailing discontinuity is a tangential discontinuity. It has a width of 2-3 ion Larmor radii. For oblique shocks, a contact discontinuity is present in the downstream plasma state. These results are of relevance to shock interactions in the very distant solar wind as well as in other energetic astrophysical situations such as solar flares.

Cargill, P. J.↗
NASA NTRS1994

Fourth order difference methods for hyperbolic IBVP's

Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

Gustafsson, Bertil↗
NASA NTRS1971

Hydromagnetic waves and discontinuities in the solar wind.

Development of a theory of hydromagnetic waves and discontinuities which is appropriate for the solar wind. The experimental evidence for the various waves, discontinuities, and some of the instabilities which are predicted by this theory is reviewed. Nearly all of the discontinuities given by the theory are shown to exist in the solar wind. These include tangential discontinuities, forward and reverse fast and slow shocks, perpendicular shocks, and Alfven shocks. Parallel shocks and contact discontinuities have not been found. A number of special cases are considered which show the basic physical properties of hydromagnetic waves in an anisotropic, multifluid, collisionless plasma. A treatment of discontinuities is presented which most resembles those of Chao (1970) and Hudson (1970). On the basis of the experimental results reviewed it is concluded that hydromagnetic theory is applicable to the solar wind.

Burlaga, L. F.↗
NASA NTRS2017

Feature Detection and Curve Fitting Using Fast Walsh Transforms for Shock Tracking: Applications

Walsh functions form an orthonormal basis set consisting of square waves. Square waves make the system well suited for detecting and representing functions with discontinuities. Given a uniform distribution of 2p cells on a one-dimensional element, it has been proven that the inner product of the Walsh Root function for group p with every polynomial of degree < or = (p - 1) across the element is identically zero. It has also been proven that the magnitude and location of a discontinuous jump, as represented by a Heaviside function, are explicitly identified by its Fast Walsh Transform (FWT) coefficients. These two proofs enable an algorithm that quickly provides a Weighted Least Squares fit to distributions across the element that include a discontinuity. The detection of a discontinuity enables analytic relations to locally describe its evolution and provide increased accuracy. Time accurate examples are provided for advection, Burgers equation, and Riemann problems (diaphragm burst) in closed tubes and de Laval nozzles. New algorithms to detect up to two C0 and/or C1 discontinuities within a single element are developed for application to the Riemann problem, in which a contact discontinuity and shock wave form after the diaphragm bursts.

Gnoffo, Peter A.↗
NASA NTRS1971

Theoretical constraints on the microscale fluctuations in the interplanetary medium.

Theoretical constraints on the character of microscale fluctuations in the solar wind are examined. The microfluctuations could be either ?discontinuities' or more smoothly varying structures. If the fluctuations are waves, they are probably mostly Alfv]en waves (or rotational discontinuities), and if they are stationary structures, they are tangential pressure balances (or tangential discontinuities). It is formally proved that a true analog of the small-amplitude MHD ?entropy wave' (or contact discontinuity) does not exist for collisionless plasmas.

Barnes, A.↗
NASA NTRS1992

The role of intermediate shocks in magnetic reconnection

The present study examines the structure of discontinuity layers associated with magnetic reconnection by numerically solving the Riemann problem for the evolution of an initial current sheet which separates two plasma regions with antiparallel magnetic field components in the z direction and a common guide magnetic field in the y direction. In the presence of a nonzero normal component of the magnetic field, the initial current sheet evolves into a system of MHD discontinuities. For the initial current sheet with a zero guide field, steady intermediate shocks, slow shocks, slow expansion waves, or contact discontinuity are observed to develop. For the current sheet with a nonzero guide field, time-dependent intermediate shocks, instead of steady intermediate shocks, are observed to bound the reconnection layer.

Lin, Y.↗
NASA NTRS1994

Motion of the heliospheric termination shock. 4: MHD effects

In this paper we generalize earlier gasdynamic analyses of the motion of the heliospheric termination shock in response to upstream disturbances (Barnes, 1993, 1994; Naidu and Barnes, 1994), to include magnetohydrodynamic (MHD) phenomena. We assume that the termination shock is a strong, perpendicular shock and that the initial upstream disturbance is a tangential discontinuity. The resulting configuration after the interaction is very similar to that in the gasdynamic models after an interaction with a contact discontinuity or interplanetary shock, and for an increase (decrease) in dynamic pressure consists of an outward (inward) propagating termination shock and an outward propagating shock (MHD rarefraction wave) that carries the signal of the disturbance into the far downstream plasma. The plasma immediately behind the new termination shock is separated from the downstream signal by a tangential discontinuity. The results of the model show that the speed of the new termination shock depends mainly on the magnitude of the change in dynamic pressure and are typically of order approximately 100 km/s, comparable to the results of the gasdynamic models.

Naidu, Kamcilla↗
NASA NTRS1994

Motion of the Heliospheric Termination Shock: MHD Effects - 4

In this paper we generalize earlier gasdynamic analyses of the motion of the heliospheric termination shock in response to upstream disturbances to include magnetohydrodynamic (MHD) phenomena. We assume that the termination shock is a strong, perpendicular shock and that the initial upstream disturbance is a tangential discontinuity. The resulting configuration after the interaction is very similar to that in the gasdynamic models after an interaction with a contact discontinuity or interplanetary shock, and for an increase (decrease) in dynamic pressure consists of an outward (inward) propagating termination shock and an outward propagating shock (MHD rarefaction wave) that carries the signal of the disturbance into the far downstream plasma. The plasma immediately behind the new termination shock is separated from the downstream signal by a tangential discontinuity. The results of the model show that the speed of the new termination shock depends mainly on the magnitude of the change in dynamic pressure and are typically of order -100 km/s, comparable to the results of the gasdynamic models.

Naidu, Kamcilla↗
NASA NTRS1993

Computing 3-D steady supersonic flow via a new Lagrangian approach

The new Lagrangian method introduced by Loh and Hui (1990) is extended for 3-D steady supersonic flow computation. Details of the conservation form, the implementation of the local Riemann solver, and the Godunov and the high resolution TVD schemes are presented. The new approach is robust yet accurate, capable of handling complicated geometry and reactions between discontinuous waves. It keeps all the advantages claimed in the 2-D method of Loh and Hui, e.g., crisp resolution for a slip surface (contact discontinuity) and automatic grid generation along the stream.

Loh, C. Y.↗
NASA NTRS2000

Visualization, Extraction and Quantification of Discontinuities in Compressible Flows

Scientific visualizations of two-dimensional compressible flow of a gas with discontinuities are presented. The numerical analogue to experimental techniques such as schlieren imaging, shadowgraphs, and interferograms are discussed. Edge detection techniques are utilized to identify the discontinuities. In particular, the zero crossing of the Laplacian of a field (usually density) is recommended for extracting the discontinuities. An algorithm to extract and quantify the discontinuities is presented. To illustrate the methods developed in the report, the example chosen is that of an unsteady interaction of a shock wave with a contact discontinuity.

Samtaney, Ravi↗
NASA NTRS1993

A new Lagrangian method for three-dimensional steady supersonic flows

In this report, the new Lagrangian method introduced by Loh and Hui is extended for three-dimensional, steady supersonic flow computation. The derivation of the conservation form and the solution of the local Riemann solver using the Godunov and the high-resolution TVD (total variation diminished) scheme is presented. This new approach is accurate and robust, capable of handling complicated geometry and interactions between discontinuous waves. Test problems show that the extended Lagrangian method retains all the advantages of the two-dimensional method (e.g., crisp resolution of a slip-surface (contact discontinuity) and automatic grid generation). In this report, we also suggest a novel three dimensional Riemann problem in which interesting and intricate flow features are present.

Loh, Ching-Yuen↗
NASA NTRS1987

ENO schemes with subcell resolution

The notion is introduced of subcell distribution, which is based on the observation that unlike point values, cell-averages of a discontinuous piecewise-smooth function contain information about the exact location of the discontinuity within the cell. Using this observation an essentially non-oscillatory (ENO) reconstruction technique is designed which is exact for cell averages of discontinuous piecewise-polynomial functions of the appropriate degree. Later on this new reconstruction technique is incorporated into Godunov-type schemes in order to produce a modification of the ENO schemes which prevents the smearing of contact discontinuities.

Harten, Ami↗
NASA NTRS1989

ENO schemes with subcell resolution

The notion is introduced of subcell distribution, which is based on the observation that unlike point values, cell-averages of a discontinuous piecewise-smooth function contain information about the exact location of the discontinuity within the cell. Using this observation, an essentially non-oscillatory (ENO) reconstruction technique is designed which is exact for cell averages of discontinuous piecewise-polynomial functions of the appropriate degree. Later on this new reconstruction technique is incorporated into Godunov-type schemes in order to produce a modification of the ENO schemes which prevents the smearing of contact discontinuities.

Harten, Ami↗